Physical Chemistry Third Edition

17.7 The Intrinsic Angular Momentum of the Electron. “Spin” 757

EXAMPLE17.10

Figure 17.12 shows the two cones of possible directions of the intrinsic angular momentum.

Find the angle between thezaxis and the intrinsic angular momentum forms+ 1 /2.

Solution

θarccos

(

h/ ̄ 2

h ̄

√

(1/2)(3/ 2

)

arccos

(

1 / 2

√

3 / 4

)

arccos(0.57735) 54. 7356 ...degrees 0. 9553166 ...radians

ms (^512)
ms (^5212)
Figure 17.12 Cones of Possible
Spin Angular Momentum Directions
for One Electron.
The total angular momentum of an electron is the vector sum of the orbital and
intrinsic angular momenta and is denoted byJ
JL+S (17.7-5)
Vector addition is described in Appendix B. Thezcomponent ofJis the algebraic sum
ofLzandSz:
JzLz+Szmh ̄+msh ̄mh ̄±

1

2

h ̄ (17.7-6)

Although it is not known what the electron’s internal structure is (if any), it is

customary to ascribe the intrinsic angular momentum to rotation of the electron about

its own axis, calling it thespin angular momentum.^6 The assumed spinning motion is

analogous to the rotation of the earth on its axis, and the orbital motion is analogous

to the earth’s revolution about the sun. We will use this spin interpretation, although

there is no guarantee that this picture is physically accurate. We will call the state for

ms+ 1 /2 the “spin up” state, and the state forms− 1 /2 the “spin down” state,

corresponding to the direction of the intrinsic angular momentum vector.

We have now twice as many possible states of electronic motion in a hydrogen-like

atom as we did before, because for every set of values of the quantum numbersn,l,

andm,mscan equal either 1/2or− 1 /2. There are two ways to include spin in our

notation. The first is to attach the value ofmsas another subscript on the orbital symbol,

replacingnlmbynlm ms. There is no need to include the value ofssince it is fixed

ats 1 /2. The orbital is now called aspin orbital. The second way, which we will

usually use, is to multiply the original orbital by aspin functionthat is calledαfor

ms+ 1 /2 andβforms− 1 /2. The original orbital is now called aspace orbital,

and the product of the space orbital and the spin function represents the spin orbital.

The two ways of writing a spin orbital are equivalent:

ψnlm,1/ 2 ψnlmα, ψnlm,− 1 / 2 ψnlmβ (17.7-7)

The spin functions are thought of as being functions of some spin coordinates that

are not explicitly represented. We define operators for the spin angular momentum that

are analogous to the orbital angular momentum operators. Since we do not have any

(^6) Superstring theory pictures the electron and other elementary particles as vibrations in 10 or 11 dimen-
sions of tiny strings with physical dimensions around 10−^35 m. For an introduction, see B. Greene,The
Elegant Universe, Vintage Books, New York, 2003.